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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 1 - Trigonometric FunctionsProblema 93
Capitolo 2, Problema 93

Use trigonometric function values of quadrantal angles to evaluate each expression. (sec 180°)² ― 3 (sin 360°)² + cos 180°

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Recall the definitions and values of trigonometric functions at quadrantal angles: 180° and 360°. For example, \( \cos 180^\circ = -1 \), \( \sin 360^\circ = 0 \), and \( \sec 180^\circ = \frac{1}{\cos 180^\circ} \).
Calculate \( \sec 180^\circ \) by taking the reciprocal of \( \cos 180^\circ \). Since \( \cos 180^\circ = -1 \), then \( \sec 180^\circ = \frac{1}{-1} = -1 \).
Square the value of \( \sec 180^\circ \) to get \( (\sec 180^\circ)^2 \). This means calculating \( (-1)^2 \).
Calculate \( (\sin 360^\circ)^2 \) by squaring the sine of 360°, which is \( 0^2 \). Then multiply this result by 3 as indicated in the expression.
Substitute all the calculated values back into the original expression \( (\sec 180^\circ)^2 - 3 (\sin 360^\circ)^2 + \cos 180^\circ \) and simplify step-by-step to evaluate the entire expression.

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Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis of the coordinate plane, typically 0°, 90°, 180°, 270°, and 360°. Their trigonometric function values are special and often take values of 0, ±1, or undefined, simplifying calculations.
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Trigonometric Function Values at Quadrantal Angles

At quadrantal angles, sine, cosine, and secant functions have specific values: for example, sin 360° = 0, cos 180° = -1, and sec 180° = 1/cos 180° = -1. Knowing these exact values helps directly evaluate expressions without approximation.
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Evaluating Expressions Using Trigonometric Identities

To evaluate expressions involving trigonometric functions, substitute known values and apply algebraic operations carefully. For example, squaring sec 180° means squaring its value, and combining terms requires attention to signs and coefficients.
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